I'd go for either "The Whisperer in Darkness" or "The Shadow over Innsmouth".
The two things I like best about Whisperer are the unreliable narrator and the sense of mounting danger. The idea of a narrator who puts the story together second-hand through recordings and letters is quite modern. The overheard recording is really effective.
Innsmouth really interests me because it hints at so much. Is there a department of the FBI that deals with the paranormal? What terrible things happen in the camps and prisons where the Deep Ones were taken? What will happen now the Navy has torpedoed a Deep One city? Also, I really like the escape from the hotel. Apparently Lovecraft didn't like the action elements, but I think they work well.
Both stories have quite an intimate, small-scale feel of one guy fighting people-sized monsters, which I find much scarier than the vague threat of Cthulhu or the like. I've thought in the past that Lovecraft tended to tell the same story over and over again, but I like the variations he brings in Innsmouth and Whisperer.
Also, I've got to mention "The Picture in the House", which isn't his best, but has this unpleasant pornographic quality that feels weirdly prescient. Not exactly "radicalised by the internet" but the concept isn't a million miles away...